Unsupported Boolean algebras and forcing

Abstract

If κ is an infinite cardinal, a complete Boolean algebra B is called κ-supported if for each sequence 〈b β : β < κ〉 of elements of B the equality ∧ α>κ ∨ β>α b β = ∨ A∈[κ]κ ∧ β∈A b β holds. Combinatorial and forcing equivalents of this property are given and compared with the other forcing related properties of Boolean algebras (distributivity, caliber, etc.). The set of regular cardinals κ for which B is not κ-supported is investigated. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.